We measure demographic parity for \(c=1\) , by the Disparate Impact (DI) measure [1]}, [2]}, [3]}: 
| P(y = 1 | a = 0) - P(y = 1 | a = 1)|

Demographic parity alone, however, is limited, in the case where the ground truth rates differ. That is, when \(P(y = c | a = 0) \ne P(y = c | a = 1)\) , where \(y\)  is the ground truth label of a random data point \(x\) .
Equality of Odds addresses disparities in the ground truth rates, and is defined as:

P(y = 1 | a = 0, y = ygt) = P(y = 1 | a = 1, y = ygt)


where \( y_{gt} \in \lbrace -1, 1\rbrace \) . Equality of Odds requires both equal false positive and true positive rates between these groups. The
Equality of Opportunity (EO)  measure is obtained by considering the equality of odds focuses on the true positive rates only :

|P(y = 1 | a = 0, y = 1) - P(y = 1 | a = 1, y = 1)|

It ensures that the classification rates are equal across groups (having different sensitive attributes).
We note that there is often a tradeoff between accuracy and fairness [4]}, since labels \(y\)  and sensitive attribute \(a\)  may be correlated.
Our framework enables the control of this tradeoff.
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